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[[Image:Complete coloring clebsch graph.svg|right|300px|thumb|
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Complete coloring of the [[Clebsch graph]] with 8 colors. Every pair of colors appears on at least one edge. No complete coloring with more colors exists: in any 9-coloring some color would appear only at one vertex, and there would not be enough neighboring vertices to cover all pairs involving that color. Therefore, the achromatic number of the Clebsch graph is 8.]]
 
In [[graph theory]], '''complete coloring''' is the opposite of [[harmonious coloring]] in the sense that it is a [[Graph coloring|vertex coloring]] in which every pair of colors appears on at least one pair of adjacent vertices. Equivalently, a complete coloring is minimal in the sense that it cannot be transformed into a proper coloring with fewer colors by merging pairs of color classes. The '''achromatic number''' ψ(G) of a graph G is the maximum number of colors possible in any complete coloring of G.
 
==Complexity theory==
Finding ψ(G) is an [[optimization problem]]. The [[decision problem]] for complete coloring can be phrased as:
 
:INSTANCE: a graph <math>G=(V,E)</math> and positive integer <math>k</math>
:QUESTION: does there exist a [[partition of a set|partition]] of <math>V</math> into <math>k</math> or more [[Disjoint sets|disjoint]] sets <math>V_1,V_2,\ldots,V_k</math> such that each <math>V_i</math> is an [[Independent set (graph theory)|independent set]] for <math>G</math> and such that for each pair of distinct sets <math>V_i,V_j,V_i \cup V_j</math> is not an independent set.
 
Determining the achromatic number is [[NP-hard]]; determining if it is greater than a given number is [[NP-complete]], as shown by Yannakakis and Gavril in 1978 by transformation from the minimum maximal matching problem.<ref name="gj">{{Citation|author = [[Michael R. Garey]] and [[David S. Johnson]] | year = 1979 | title = Computers and Intractability: A Guide to the Theory of NP-Completeness | publisher = W.H. Freeman | isbn = 0-7167-1045-5}} A1.1: GT5, pg.191.</ref>
 
Note that any coloring of a graph with the minimum number of colors must be a complete coloring, so minimizing the number of colors in a complete coloring is just a restatement of the standard [[graph coloring]] problem.
 
==Algorithms==
For any fixed ''k'', it is possible to determine whether the achromatic number of a given graph is at least ''k'', in linear time.<ref name="fhhm"/>
 
The optimization problem permits approximation and is approximable within a <math>O\left(|V|/\sqrt{\log |V|}\right)</math> [[approximation ratio]].<ref name="cv">{{citation|last1=Chaudhary|first1=Amitabh|last2=Vishwanathan|first2=Sundar|title=Approximation algorithms for the achromatic number|journal=Journal of Algorithms|volume=41|year=2001|issue=2|pages=404–416|doi=10.1006/jagm.2001.1192}}.</ref>
 
==Special classes of graphs==
The NP-completeness of the achromatic number problem holds also for some special classes of graphs:
[[bipartite graph]]s,<ref name="fhhm">{{citation|last1=Farber|first1=M.|last2=Hahn|first2=G.|last3=Hell|first3=P.|last4=Miller|first4=D. J.|title=Concerning the achromatic number of graphs|journal=Journal of Combinatorial Theory, Series B|volume=40|issue=1|year=1986|pages=21–39|doi=10.1016/0095-8956(86)90062-6}}.</ref>
[[graph complement|complements]] of [[bipartite graph]]s (that is, graphs having no independent set of more than two vertices),<ref name="gj"/> [[cograph]]s and [[interval graph]]s,<ref>{{citation|last=Bodlaender|first=H.|authorlink=Hans L. Bodlaender|title=Achromatic number is NP-complete for cographs and interval graphs|journal=Inform. Proc. Lett.|volume=31|year=1989|issue=3|pages=135–138|doi=10.1016/0020-0190(89)90221-4}}.</ref> and even for trees.<ref>{{citation|last1=Manlove|first1=D.|last2=McDiarmid|first2=C.|title=The complexity of harmonious coloring for trees|journal=Discrete Applied Mathematics|volume=57|year=1995|issue=2-3|pages=133–144|doi=10.1016/0166-218X(94)00100-R}}.</ref>
 
For complements of trees, the achromatic number can be computed in polynomial time.<ref>{{citation|last1=Yannakakis|first1=M.|last2=Gavril|first2=F.|title=Edge dominating sets in graphs|journal=SIAM Journal on Applied Mathematics|volume=38|year=1980|issue=3|pages=364–372|doi=10.1137/0138030}}.</ref> For trees, it can be approximated to within a constant factor.<ref name="cv"/>
 
The achromatic number of an ''n''-dimensional [[hypercube graph]] is known to be proportional to <math>\sqrt{n2^n}</math>, but the constant of proportionality is not known precisely.<ref>{{citation|last=Roichman|first=Y.|title= On the Achromatic Number of Hypercubes|journal=Journal of Combinatorial Theory, Series B|volume=79|issue=2|year=2000|pages=177–182|doi=10.1006/jctb.2000.1955}}.</ref>
 
==References==
{{reflist|2}}
 
==External links==
* [http://www.nada.kth.se/~viggo/wwwcompendium/node17.html A compendium of NP optimization problems]
* [http://www.maths.dundee.ac.uk/~kedwards/biblio.html ''A Bibliography of Harmonious Colourings and Achromatic Number''] by Keith Edwards
 
{{DEFAULTSORT:Complete Coloring}}
[[Category:Graph coloring]]

Revision as of 01:16, 9 February 2014

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